Ratios and proportions and how to solve them

Let's talk about ratios and proportions. When we talk about the
speed of a car or an airplane we measure it in miles per hour. This
is called a rate and is a type of ratio. A ratio is a way to
compare two quantities by using division as in miles per hour where
we compare miles and hours.

A ratio can be written in three different ways and all are read
as "the ratio of x to y"

A proportion on the other hand is an equation that says that two
ratios are equivalent. For instance if one package of cookie mix
results in 20 cookies than that would be the same as to say that
two packages will result in 40 cookies.

A proportion is read as "x is to y as z is to w"

If one number in a proportion is unknown you can find that
number by solving the proportion.

Example:

You know that to make 20 pancakes you have to use 2 eggs. How
many eggs are needed to make 100 pancakes?

Eggs

pancakes

Small amount

2

20

Large amount

x

100

If we write the unknown number in the nominator then we can
solve this as any other equation

Multiply both sides with 100

If the unknown number is in the denominator we can use another
method that involves the cross product. The cross product is the
product of the numerator of one of the ratios and the denominator
of the second ratio. The cross products of a proportion is always
equal

If we again use the example with the cookie mix used above

It is said that in a proportion if

If you look at a map it always tells you in one of the corners
that 1 inch of the map correspond to a much bigger distance in
reality. This is called a scaling. We often use scaling in order to
depict various objects. Scaling involves recreating a model of the
object and sharing its proportions, but where the size differs. One
may scale up (enlarge) or scale down (reduce). For example,
the scale of 1:4 represents a fourth. Thus any measurement we see
in the model would be 1/4 of the real measurement. If we wish to
calculate the inverse, where we have a 20ft high wall and wish to
reproduce it in the scale of 1:4, we simply calculate:

In a scale model of 1:X where X is a constant, all measurements
become 1/X - of the real measurement. The same mathematics applies
when we wish to enlarge. Depicting something in the scale of 2:1
all measurements then become twice as large as in reality. We
divide by 2 when we wish to find the actual measurement.